Truly nonlinear oscillations : harmonic balance, parameter expansions, iteration, and averaging methods
Approximation theory Nonlinear oscillations e-böcker
World Scientific
2010
EISBN 9789814291668
1. Background and general comments. 1.1. Truly nonlinear functions. 1.2. Truly nonlinear oscillators. 1.3. General remarks. 1.4. Scaling and dimensionless form of differential equations. 1.5. Exactly solvable TNL oscillators. 1.6. Overview of TNL oscillator methods. 1.7. Discussion.
2. Establishing periodicity. 2.1. Phase-space. 2.2. Application of phase-space methods. 2.3. Dissipative systems : energy methods. 2.4. Resume.
3. Harmonic balance. 3.1. Direct harmonic balance : methodology. 3.2. Worked examples. 3.3. Rational approximations. 3.4. Worked examples. 3.5. Third-order equations. 3.6. Resume.
4. Parameter expansions. 4.1. Introduction. 4.2. Worked examples. 4.3. Discussion.
5. Iteration methods. 5.1. General methodology. 5.2. Worked examples : direct iteration. 5.3. Worked examples : extended iteration. 5.4. Discussion.
6. Averaging methods. 6.1. Elementary TNL averaging methods. 6.2. Worked examples. 6.3. Cveticanin's averaging method. 6.4. Worked examples. 6.5. Chronology of averaging methods. 6.6. Comments.
7. Comparative analysis. 7.1. Purpose. 7.2. x + x[symbol] = 0. 7.3. x + x[symbol] = 0. 7.4. x + x[symbol] = -2[symbol]. 7.5. x + x[symbol] = -2[symbol]. 7.6. x + x[symbol] = [symbol]. 7.7. x + x[symbol] = [symbol]. 7.8. General comments and calculation strategies. 7.9. Research problems.
This unique book provides a concise presentation of many of the fundamental strategies for calculating approximations to the oscillatory solutions of "truly nonlinear" (TNL) oscillator equations. The volume gives a general overview of the author's work on harmonic balance, iteration and combined linearization-averaging methods. However, full discussions are also presented on parameter expansion procedures and a first-order averaging technique for TNL oscillators. The calculational basis of each method is clarified by applying them to a set of standard TNL oscillator equations. This allows a direct comparison to be made among the various methods. The book is self-contained and therefore suitable for both classroom use and self-study by students and professionals who desire to learn, understand, and apply these technique to the field of nonlinear oscillations.
2. Establishing periodicity. 2.1. Phase-space. 2.2. Application of phase-space methods. 2.3. Dissipative systems : energy methods. 2.4. Resume.
3. Harmonic balance. 3.1. Direct harmonic balance : methodology. 3.2. Worked examples. 3.3. Rational approximations. 3.4. Worked examples. 3.5. Third-order equations. 3.6. Resume.
4. Parameter expansions. 4.1. Introduction. 4.2. Worked examples. 4.3. Discussion.
5. Iteration methods. 5.1. General methodology. 5.2. Worked examples : direct iteration. 5.3. Worked examples : extended iteration. 5.4. Discussion.
6. Averaging methods. 6.1. Elementary TNL averaging methods. 6.2. Worked examples. 6.3. Cveticanin's averaging method. 6.4. Worked examples. 6.5. Chronology of averaging methods. 6.6. Comments.
7. Comparative analysis. 7.1. Purpose. 7.2. x + x[symbol] = 0. 7.3. x + x[symbol] = 0. 7.4. x + x[symbol] = -2[symbol]. 7.5. x + x[symbol] = -2[symbol]. 7.6. x + x[symbol] = [symbol]. 7.7. x + x[symbol] = [symbol]. 7.8. General comments and calculation strategies. 7.9. Research problems.
This unique book provides a concise presentation of many of the fundamental strategies for calculating approximations to the oscillatory solutions of "truly nonlinear" (TNL) oscillator equations. The volume gives a general overview of the author's work on harmonic balance, iteration and combined linearization-averaging methods. However, full discussions are also presented on parameter expansion procedures and a first-order averaging technique for TNL oscillators. The calculational basis of each method is clarified by applying them to a set of standard TNL oscillator equations. This allows a direct comparison to be made among the various methods. The book is self-contained and therefore suitable for both classroom use and self-study by students and professionals who desire to learn, understand, and apply these technique to the field of nonlinear oscillations.
